Let \(V_n\in \mathcal {V}\)  be a minimizing sequence for the optimization problem (REF ). Then, \(v_n=\big (\Psi (V_n)\big )^{1/p}\)  is a bounded sequence in \(L^p(D)\)  and so, up to a subsequence, \(v_n\)  converges weakly in \(L^p(D)\)  to some function \(v\) . We prove that the potential \(V=\Psi ^{-1}(v^p)\)  is a solution to (REF ). By its definition we have \(V\in \mathcal {V}\)  and so it remains to prove that
\(\int _D gR_V(f)\,dx\le \liminf _n\int _D gR_{V_n}(f)\,dx\;.\) 
Since the \(\gamma \) -convergence is compact, we may assume that, up to a subsequence, \(V_n\)  \(\gamma \) -converges to a capacitary measure \(\mu \in \overline{\mathcal {V}}\) , which implies
\(\int _D gR_\mu (f)\,dx=\lim _n\int _D gR_{V_n}(f)\,dx\;.\) 
Therefore, it remains only to prove the inequality
\(\int _D gR_V(f)\,dx\le \int _D gR_\mu (f)\,dx\;.\) 
By the definition of \(\gamma \) -convergence, we have that for any \(u\in H^1_0(D)\) , there is a sequence \(u_n\in H^1_0(D)\)  which converges to \(u\)  in \(L^2(D)\)  and is such that
\(\begin{split}\int _D|\nabla u|^2\,dx+\int _D u^2\,d\mu &=\lim _{n\rightarrow \infty }\int _D|\nabla u_n|^2\,dx+\int _D u_n^2 V_n\,dx\\&=\lim _{n\rightarrow \infty }\int _D|\nabla u_n|^2\,dx+\int _D u_n^2\Psi ^{-1}(v_n^p)\,dx\\&\ge \int _D|\nabla u|^2\,dx+\int _D u^2\Psi ^{-1}(v^p)\,dx\\&=\int _D|\nabla u|^2\,dx+\int _D u^2V\,dx,\end{split}\) 
where the inequality above is due to the strong-weak lower semicontinuity of integral functionals (see for instance [1]}), which follows by the assumptions made on the function \(\Psi \) . Thus, for any \(u\in H^1_0(D)\) , we have
\(\int _D u^2\,d\mu \ge \int _D u^2V\,dx,\) 
which gives \(V\le \mu \) . The inequality (REF ) now follows by Lemma REF .
